佩雷尔曼关于庞加莱猜想的论文0303109.pdf

arXiv:math.DG/0303109 v1 10 Mar 2003 Ricci fl ow with surgery on three-manifolds Grisha Perelman August 21, 2006 This is a technical paper, which is a continuation of I. Here we verify most of the assertions, made in I, 13; the exceptions are (1) the statement that a 3-manifold which collapses with local lower bound for sectional curvature is a graph manifold - this is deferred to a separate paper, as the proof has nothing to do with the Ricci fl ow, and (2) the claim about the lower bound for the volumes of the maximal horns and the smoothness of the solution from some time on, which turned out to be unjustifi ed, and, on the other hand, irrelevant for the other conclusions. The Ricci fl ow with surgery was considered by Hamilton H 5,4,5; unfortu- nately, his argument, as written, contains an unjustifi ed statement (RMAX= , on page 62, lines 7-10 from the bottom), which I was unable to fi x. Our approach is somewhat diff erent, and is aimed at eventually constructing a canonical Ricci fl ow, defi ned on a largest possible subset of space-time, - a goal, that has not been achieved yet in the present work. For this reason, we consider two scale bounds: the cutoff radius h, which is the radius of the necks, where the surg- eries are performed, and the much larger radius r, such that the solution on the scales less than r has standard geometry. The point is to make h arbitrarily small while keeping r bounded away from zero. Notation and terminology B(x,t,r) denotes the open metric ball of radius r, with respect to the metric at time t, centered at x. P(x,t,r,t) denotes a parabolic neighborhood, that is the set of all points (x,t) with x B(x,t,r) and t t,t + t or t t + t,t, depending on the sign of t. A ball B(x,t,1r) is called an -neck, if, after scaling the metric with factor r2, it is -close to the standard neck S2I, with the product metric, where S2 has constant scalar curvature one, and I has length 21; here -close refers to CNtopology, with N 1. A parabolic neighborhood P(x,t,1r,r2) is called a strong -neck, if, after scaling with factor r2, it is -close to the evolving standard neck, which at each St.Petersburg branch of Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia. Email: perelmanpdmi.ras.ru or 1 time t 1,0 has length 21and scalar curvature (1 t)1. A metric on S2 I, such that each point is contained in some -neck, is called an -tube, or an -horn, or a double -horn, if the scalar curvature stays bounded on both ends, stays bounded on one end and tends to infi nity on the other, and tends to infi nity on both ends, respectively. A metric on B3or RP3 B3, such that each point outside some compact subset is contained in an -neck, is called an -cap or a capped -horn, if the scalar curvature stays bounded or tends to infi nity on the end, respectively. We denote by a fi xed small positive constant. In contrast, denotes a positive quantity, which is supposed to be as small as needed in each particular argument. 1Ancient solutions with bounded entropy 1.1 In this section we review some of the results, proved or quoted in I,11, correcting a few inaccuracies. We consider smooth solutions gij(t) to the Ricci fl ow on oriented 3-manifold M, defi ned for 0; such solutions will be called ancient -solutions for short. By Theorem I.11.7, the set of all such solutions with fi xed is compact modulo scaling, that is from any sequence of such solutions (M,g ij(t) and points (x,0) with R(x,0) = 1, we can extract a smoothly (pointed) convergent subsequence, and the limit (M,gij(t) belongs to the same class of solutions. (The assumption in I.11.7. that M be noncompact was clearly redundant, as it was not used in the proof. Note also that M need not have the same topology as M.) Moreover, according to Proposition I.11.2, the scalings of any ancient -solution gij(t) with factors (t)1about appropriate points converge along a subsequence of t to a non-fl at gradient shrinking soliton, which will be called an asymptotic soliton of the ancient solution. If the sectional curvature of this asymptotic soliton is not strictly positive, then by Hamiltons strong maximum principle it admits local metric splitting, and it is easy to see that in this case the soliton is either the round infi nite cylinder, or its Z2quotient, containing one-sided projective plane. If the curvature is strictly positive and the soliton is compact, then it has to be a metric quotient of the round 3-sphere, by H 1. The noncompact case is ruled out below. 1.2 Lemma. There is no (complete oriented 3-dimensional) noncompact -noncollapsed gradient shrinking soliton with bounded positive sectional curva- ture. Proof. A gradient shrinking soliton gij (t), t 1), with two spherical caps, smoothly attached to its boundary components. By H 1 we know that the fl ow shrinks such a metric to a point in time, comparable to one (because both the lower bound for scalar curvature and the upper bound for sectional curvature are comparable to one) , and after 3 normalization, the fl ow converges to the round 3-sphere. Scale the initial metric and choose the time parameter in such a way that the fl ow starts at time t0= t0(L) 0 independent of L. We also claim that t0(L) as L . Indeed, the Harnack inequality of Hamilton H 3 implies that Rt R t0t, hence R 2(1t0) tt0 for t 1, and then the distance change estimate d dtdistt(x,y) const pR max(t) from H 2,17 implies that the diameter of gij(t0) does not exceed const t0, which is less than Lt0unless t0is large enough. Thus, a subsequence of our solutions with L converges to an ancient -solution on S3, whose asymptotic soliton can not be anything but the cylinder. 1.5 The important conclusion from the classifi cation above and the proof of Proposition I.11.2 is that there exists 0 0, such that every ancient -solution is either 0-solution, or a metric quotient of the round sphere. Therefore, the compactness theorem I.11.7 implies the existence of a universal constant , such that at each point of every ancient -solution we have estimates |R| R 3 2,|Rt| 0 one can fi nd C1,2= C1,2(), such that for each point (x,t) in every ancient -solution there is a radius r,0 r C1R(x,t) 1 2, and a neighborhood B,B(x,t,r) B B(x,t,2r), which falls into one of the four categories: (a) B is a strong -neck (more precisely, the slice of a strong -neck at its maximal time), or (b) B is an -cap, or (c) B is a closed manifold, diff eomorphic to S3or RP3, or (d) B is a closed manifold of constant positive sectional curvature; furthermore, the scalar curvature in B at time t is between C1 2 R(x,t) and C2R(x,t), its volume in cases (a),(b),(c) is greater than C1 2 R(x,t) 3 2, and in case (c) the sectional curvature in B at time t is greater than C1 2 R(x,t). 2The standard solution Consider a rotationally symmetric metric on R3with nonnegative sectional cur- vature, which splits at infi nity as the metric product of a ray and the round 2-sphere of scalar curvature one. At this point we make some choice for the metric on the cap, and will refer to it as the standard cap; unfortunately, the most obvious choice, the round hemisphere, does not fi t, because the metric on R3would not be smooth enough, however we can make our choice as close to it as we like. Take such a metric on R3as the initial data for a solution gij(t) to the Ricci fl ow on some time interval 0,T), which has bounded curvature for each t 0,T). Claim 1. The solution is rotationally symmetric for all t. 4 Indeed, if ui is a vector fi eld evolving by ui t= ui+ Rijuj, then vij = iuj evolves by (vij)t= vij+ 2Rikjlvkl Rikvkj Rkjvik. Therefore, if uiwas a Killing fi eld at time zero, it would stay Killing by the maximum principle. It is also clear that the center of the cap, that is the unique maximum point for the Busemann function, and the unique point, where all the Killing fi elds vanish, retains these properties, and the gradient of the distance function from this point stays orthogonal to all the Killing fi elds. Thus, the rotational symmetry is preserved. Claim 2. The solution converges at infi nity to the standard solution on the round infi nite cylinder of scalar curvature one. In particular, T 1. Claim 3. The solution is unique. Indeed, using Claim 1, we can reduce the linearized Ricci fl ow equation to the system of two equations on (,+) of the following type ft= f+ a1f+ b1g+ c1f + d1g,gt= a2f+ b2g+ c2f + d2g, where the coeffi cients and their derivatives are bounded, and the unknowns f,g and their derivatives tend to zero at infi nity by Claim 2. So we get uniqueness by looking at the integrals RA A(f 2 + g2) as A . Claim 4. The solution can be extended to the time interval 0,1). Indeed, we can obtain our solution as a limit of the solutions on S3, starting from the round cylinder S2 I of length L and scalar curvature one, with two caps attached; the limit is taken about the center p of one of the caps, L . Assume that our solution goes singular at some time T 1. Take T1 T very close to T,T T1 0, we can fi nd L, D , depending on and T1, such that for any point x at distance D from p at time zero, in the solution with L L, the ball B(x,T1,1) is -close to the corresponding ball in the round cylinder of scalar curvature (1 T1)1. We can also fi nd r = r(,T), independent of T1, such that the ball B(x,T1,r) is -close to the corresponding euclidean ball.Now we can apply Theorem I.10.1 and get a uniform estimate on the curvature at x as t T, provided that T T1 2r(,T)2. Therefore, the t T limit of our limit solution on the capped infi nite cylinder will be smooth near x. Thus, this limit will be a positively curved space with a conical point.However, this leads to a contradiction via a blow-up argument; see the end of the proof of the Claim 2 in I.12.1. The solution constructed above will be called the standard solution. Claim 5. The standard solution satisfi es the conclusions of 1.5 , for an appropriate choice of , ,C1(),C2(), except that the -neck neighborhood need not be strong; more precisely, we claim that if (x,t) has neither an -cap neigh- borhood as in 1.5(b), nor a strong -neck neighborhood as in 1.5(a), then x is not in B(p,0,1),t 0 I, 4.Then by Theorem I.12.1 and the conclusions of 1.5 we can fi nd r = r() 0, such that each point (x,t) with R(x,t) r2 satisfi es the estimates (1.3) and has a neighborhood, which is either an -neck, or an -cap, or a closed positively curved manifold. In the latter case the solution becomes extinct at time T, so we dont need to consider it any more. If this case does not occur, then let denote the set of all points in M, where curvature stays bounded as t T. The estimates (1.3) imply that is open and that R(x,t) as t T for each x M. If is empty, then the solution becomes extinct at time T and it is entirely covered by -necks and caps shortly before that time, so it is easy to see that M is diff eomorphic to either S3, or RP3, or S2 S1, or RP3 RP3. Otherwise, if is not empty, we may (using the local derivative estimates due to W.-X.Shi, see H 2,13) consider a smooth metric gijon , which is the limit of gij(t) as t T. Let for some 0 which is admissible in sections 1,2. In this section we consider only solutions to the Ricci fl ow with surgery, which satisfy the following a priori assumptions: (pinching) There exists a function , decreasing to zero at infi nity, such that Rm (R)R, (canonical neighborhood) There exists r 0, such that every point where scalar curvature is at least r2has a neighborhood, satisfying the conclusions of 1.5. (In particular, this means that if in case (a) the neighborhood in ques- tion is B(x0,t0,1r0 ), then the solution is required to be defi ned in the whole P(x0,t0,1r0,r2 0); however, this does not rule out a surgery in the time in- terval (t0 r2 0,t0 ), that occurs suffi ciently far from x0.) Recall that from the pinching estimate of Ivey and Hamilton, and Theorem I.12.1, we know that the a priori assumptions above hold for a smooth solution on any fi nite time interval. For Ricci fl ow with surgery they will be justifi ed in the next section. 4.2 Claim 1. Suppose we have a solution to the Ricci fl ow with surgery, sat- isfying the canonical neighborhood assumption, and let Q = R(x0,t0)+r2. Then we have estimate R(x,t) 8Q for those (x,t) P(x0,t0, 1 2 1Q1 2,1 8 1Q1), for which the solution is defi ned. 7 Indeed, this follows from estimates (1.3). Claim 2. For any A r2for each y , and Q0= R(x0,t0) is so large that (Q0) 10C2R(x0,t0). Then distt0(x0,z) AQ 1 2 0 whenever R(x,t0) QQ0. The proof is exactly the same as for Claim 2 in Theorem I.12.1; in the very end of it, when we get a piece of a non-fl at metric cone as a blow-up limit, we get a contradiction to the canonical neighborhood assumption, because the canonical neighborhoods of types other than (a) are not close to a piece of metric cone, and type (a) is ruled out by the strong maximum principle, since the -neck in question is strong. 4.3 Suppose we have a solution to the Ricci fl ow with surgery, satisfying our a priori assumptions, defi ned on 0,T), and going singular at time T. Choose a small 0 and let = r. As in the previous section, consider the limit (, gij) of our solution as t T, and the corresponding compact set . Lemma. There exists a radius h,0 . Now we can continue our solution until it becomes singular for the next time. Note that after the surgery the manifold may become disconnected; in this case, each component should be dealt with separately. Furthermore, let us agree to declare extinct every component which is -close to a metric quotient of the round sphere; that allows to exclude such components from the list of canonical neighborhoods. Now since every surgery reduces the volume by at least h3, the sequence of surgery times is discrete, and, taking for granted the a priori assumptions, we can continue our solution indefi nitely, not ruling out the possibility that it may become extinct at some fi nite time. 4.5 In order to justify the canonical neighborhood assumption in the next section, we need to check several assertions. Lemma. For any A ,0 1, one can fi nd = (A,) with the following property. Suppose we have a solution to the Ricci fl ow with -cutoff , satisfying the a priori assumptions on 0,T, with . Suppose we have a surgery at time T0 (0,T), let p correspond to the center of the standard cap, and let T1= min(T,T0+ h2). Then e